SPEAKER:
    Dimitri Gioev, Courant Institute and University of Rochester
TITLE:
    Universality at the edge of the spectrum
    for unitary, orthogonal and symplectic ensembles
    of random matrices
ABSTRACT:
    In recent joint work with Percy Deift [DG]
    we proved universality in the bulk of the spectrum
    for orthogonal and symplectic ensembles
    in the scaling limit for a class of weights
    w(x)=exp(-V(x)) where V is a polynomial,
    V(x)= c_{2m} x^{2m} + ..., c_{2m}>0.
    (For unitary ensembles, universality in the bulk
    for the same class of weights was established
    by Deift-Kriecherbauer-McLaughlin-Venakides-Zhou [DKMVZ].)

    In this talk we will present the proof of universality
    at the edge of the spectrum for all the three
    invariant ensembles for the same class of weights.
    For the unitary case, the proof relies on [DKMVZ].
    In the orthogonal and symplectic cases,
    the proof depends on [DG], as well as on [DKMVZ].
    The starting point in the analysis in [DG],
    and also in the present work,
    is the representation of the orthogonal and symplectic
    correlation kernels as a sum of the unitary (Christoffel-Darboux)
    kernel together with a correction which is expressed
    in terms of the orthogonal polynomials corresponding
    to the given weight.
    This representation is due to Widom
    (whose work [W] depends in turn on earlier work of Tracy and Widom).
    It was shown in [DG] that Widom's correction term
    washes out in the bulk scaling limit.
    A new feature of the scaling limit at the edge
    is that Widom's correction term has the same order
    as the Christoffel-Darboux term,
    however its contribution to the final answer
    is the same for an arbitrary weight in the class described above
    as for the case of the Gaussian weight (V(x)=x^2).

    This is a joint work with P.Deift.